realPTsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

💖 realPTsolve(3) — the LAPACK glow-up 💖

okay bestie, pull up a chair, grab a boba 🧋, because we're about to get intimately familiar with realPTsolve, aka the real solve driver functions for PT matrices, straight out of the LAPACK manual (manual section 3, if you're keeping track 👑). This is the group chat of routines that solve A * X = B when A is a real symmetric positive definite tridiagonal matrix. Slay.

✨ NAME ✨

realPTsolve — real. That's it. That's the name. Short, iconic, no notes. 💅

🦋 SYNOPSIS 🦋

Functions

  • subroutine sptsv (N, NRHS, D, E, B, LDB, INFO) — SPTSV computes the solution to system of linear equations A * X = B for PT matrices 🌸
  • subroutine sptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, INFO) — SPTSVX computes the solution to system of linear equations A * X = B for PT matrices 💜

🌈 Detailed Description 🌈

This is the group of real solve driver functions for PT matrices. They're the main characters of this page, fr fr. 🔥

💅 Function Documentation 💅

🎀 subroutine sptsv 🎀

subroutine sptsv (integer N, integer NRHS, real, dimension( * ) D, real,
    dimension( * ) E, real, dimension( ldb, * ) B, integer LDB, integer
    INFO)

SPTSV computes the solution to system of linear equations A * X = B for PT matrices. It's giving solver energy.

Purpose:

SPTSV computes the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices.

A is factored as A = L*D*L**T, and the factored form of A is then used to solve the system of equations. Lowkey elegant, highkey efficient. ✨

Parameters

  • N
    • N is INTEGER
    • The order of the matrix A. N >= 0.
  • NRHS
    • NRHS is INTEGER
    • The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • D
    • D is REAL array, dimension (N)
    • On entry, the n diagonal elements of the tridiagonal matrix A. On exit, the n diagonal elements of the diagonal matrix D from the factorization A = L*D*L**T.
  • E
    • E is REAL array, dimension (N-1)
    • On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A. On exit, the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. (E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**T*D*U factorization of A.)
  • B
    • B is REAL array, dimension (LDB,NRHS)
    • On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. glow-up complete 💫
  • LDB
    • LDB is INTEGER
    • The leading dimension of the array B. LDB >= max(1,N).
  • INFO
    • INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the leading minor of order i is not positive definite, and the solution has not been computed. The factorization has not been completed unless i = N.
not the leading minor catching strays 😭 if you get a positive INFO, the vibes are off and the solution just... isn't there. no solution for you, bestie.

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🌸 subroutine sptsvx 🌸

subroutine sptsvx (character FACT, integer N, integer NRHS, real, dimension(
    * ) D, real, dimension( * ) E, real, dimension( * ) DF, real, dimension(
    * ) EF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx,
    * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real,
    dimension( * ) BERR, real, dimension( * ) WORK, integer INFO)

SPTSVX computes the solution to system of linear equations A * X = B for PT matrices. She does everything SPTSV does plus error bounds and a condition estimate — she's the overachiever of the family. 👑

Purpose:

SPTSVX uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. Obsessed. 🫶

Description:

The following steps are performed:

  1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U**T*D*U.
  2. If the leading i-by-i principal minor is not positive definite, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
  3. The system of equations is solved for X using the factored form of A.
  4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Parameters

  • FACT
    • FACT is CHARACTER*1
    • Specifies whether or not the factored form of A has been supplied on entry.
    • = 'F': On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified.
    • = 'N': The matrix A will be copied to DF and EF and factored.
  • N
    • N is INTEGER
    • The order of the matrix A. N >= 0.
  • NRHS
    • NRHS is INTEGER
    • The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • D
    • D is REAL array, dimension (N)
    • The n diagonal elements of the tridiagonal matrix A.
  • E
    • E is REAL array, dimension (N-1)
    • The (n-1) subdiagonal elements of the tridiagonal matrix A.
  • DF
    • DF is REAL array, dimension (N)
    • If FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A. If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A.
  • EF
    • EF is REAL array, dimension (N-1)
    • If FACT = 'F', then EF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. If FACT = 'N', then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A.
  • B
    • B is REAL array, dimension (LDB,NRHS)
    • The N-by-NRHS right hand side matrix B.
  • LDB
    • LDB is INTEGER
    • The leading dimension of the array B. LDB >= max(1,N).
  • X
    • X is REAL array, dimension (LDX,NRHS)
    • If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
  • LDX
    • LDX is INTEGER
    • The leading dimension of the array X. LDX >= max(1,N).
  • RCOND
    • RCOND is REAL
    • The reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.
  • FERR
    • FERR is REAL array, dimension (NRHS)
    • The forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j).
  • BERR
    • BERR is REAL array, dimension (NRHS)
    • The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
  • WORK
    • WORK is REAL array, dimension (2*N)
  • INFO
    • INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, and i is
      • <= N: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned.
      • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.
bestie listen 😭 if RCOND is giving zero, the matrix is singular to working precision and that's a whole situation. and if INFO = N+1? the matrix is singular to working precision but the routine is like "we ball" and computes the solution anyway, because sometimes the computed solution is more accurate than RCOND would have you believe. the audacity. the confidence. we love to see it. 💀✨

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code. (not a human, just a robot doing its little thing, and honestly? it ate. 🤖💖)

and that's the whole page, bestie!! every flag, every array dimension, every INFO code, all of it, no cap. go forth and solve those tridiagonal systems. you're the main character of linear algebra now. 💜

mwah 💋 this was LAPACK Version 3.11.0, dated Sun Nov 27 2022 — big love to the LAPACK girlies, and to the realPTsolve(3) manual page for having us. periodt. 🎀✨🦋

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